Question
Fix a positive integer $k$. You have $r$ very loyal rats that are willing to taste cups of wine for you! Exactly one of the $n$ cups has poison in it at random, which kills rats that drink it one hour after sipping. Otherwise, the rat drinks wine and feels a little tipsy, but it powers through and is fine. The rats are busy, so they can only drink cups within a one-hour period. Find the maximum value of $n$ such that the poisonous cup can be identified with probability at least $1/k$ when $r = 10$ and $k = 5$. Assume the sipping process is instantaneous.